Second-Order ODEs and Linear Systems

A second-order differential equation involves the second derivative of the unknown. In the linear case it reads

and it is homogeneous if $f\equiv 0$, complete otherwise. We will almost always work with constant coefficients: $a,b,c$ numbers.

Why second order is special. Newton's second law $F=ma=m\,x''$ is intrinsically second-order: force determines acceleration, not position or velocity. That is why knowing only the initial position is not enough to predict the future: the initial velocity is also needed.

The Cauchy problem indeed requires two initial conditions:

Geometrically: you fix both the starting point and the starting slope. For linear equations with continuous coefficients, existence and uniqueness hold on the whole interval where $a(x)\neq0$.

General strategy. We always solve in two stages: first the associated homogeneous equation (giving the "structure" of the free system), then a particular solution of the complete one (the response to the external forcing). Their sum is the general solution.

For linear equations the superposition principle holds: linear combinations of homogeneous solutions are again solutions. The solution set of the second-order homogeneous equation is a 2-dimensional vector space: finding two independent solutions $y_1,y_2$ generates all of them.

The general solution of the complete equation is

with $y_P$ any particular solution. The two constants $C_1,C_2$ are then fixed by the two initial conditions.

When are $y_1,y_2$ truly independent? Measured by the Wronskian:

If $W\neq 0$ (even at a single point, for solutions of the same linear ODE) then $y_1,y_2$ are linearly independent and form a fundamental system. If $W\equiv 0$ they are proportional and cannot span the solution space.

Why it matters. The Wronskian guarantees that, given $y(x_0)$ and $y'(x_0)$, the system for $C_1,C_2$ is uniquely solvable: its coefficient matrix has determinant $W(x_0)\neq 0$.

For the constant-coefficient homogeneous equation $ay''+by'+cy=0$ we try exponential solutions $y=e^{\lambda x}$. Substituting ($y'=\lambda e^{\lambda x}$, $y''=\lambda^2 e^{\lambda x}$) and dividing by $e^{\lambda x}\neq 0$ gives the characteristic equation:

The sign of the discriminant $\Delta$ yields three scenarios, corresponding to three distinct physical behaviours:

• $\Delta>0$ (distinct real roots $\lambda_1\neq\lambda_2$): $y_H = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$. Sum of two exponentials — overdamped behaviour, no oscillation.
• $\Delta=0$ (double root $\lambda$): one exponential is not enough; the second independent solution is $x\,e^{\lambda x}$, so $y_H = (C_1 + C_2 x)\,e^{\lambda x}$. This is critical damping.
• $\Delta<0$ (complex roots $\lambda=\alpha\pm i\beta$): using Euler's formula, $y_H = e^{\alpha x}\big(C_1\cos\beta x + C_2\sin\beta x\big)$. Oscillation of angular frequency $\beta$ with amplitude modulated by $e^{\alpha x}$ (decaying if $\alpha<0$).

Where does $xe^{\lambda x}$ come from? When $\Delta=0$ the two roots "collapse". Taking the limit $\lambda_2\to\lambda_1$ of the combination $(e^{\lambda_2 x}-e^{\lambda_1 x})/(\lambda_2-\lambda_1)$ yields exactly $xe^{\lambda x}$: it is the derivative of $e^{\lambda x}$ with respect to $\lambda$.

When the forcing term $f(x)$ is "simple" (polynomials, exponentials, sines/cosines and their products), guess the form of $y_P$ with coefficients to be determined. The idea: the response has the same "shape" as the forcing.

The resonance rule. If the trial is already a homogeneous solution (i.e. $k$ or $i\omega$ is a characteristic root), multiply it by $x$ (or $x^2$ if the root is double). This is exactly what produces the linear growth in resonance.

Example: for $y''-3y'+2y=e^{3x}$ try $y_P=Ae^{3x}$; substituting, $(9-9+2)Ae^{3x}=e^{3x}\Rightarrow A=1/2$. If instead the forcing were $e^{x}$ ($\lambda=1$ is a root), one would try $y_P=Axe^{x}$.

This is the general method for the particular solution: it works for any $f(x)$, even when undetermined coefficients fail (e.g. $f(x)=\tan x$). Start from a fundamental system $y_1,y_2$ of the homogeneous equation and seek

letting the constants "vary". Imposing $u_1'y_1+u_2'y_2=0$ (a convenient constraint) and substituting into the equation yields:

where $W=W(y_1,y_2)$ is the Wronskian and $a$ the coefficient of $y''$. Integrate to obtain $u_1,u_2$ and hence $y_P$.

When to use it. Undetermined coefficients is faster for standard forcing terms; variation of parameters is the universal safety net. Conceptually it shows the particular solution is an integral of the forcing "weighted" by the homogeneous solutions — the idea behind the Green's function.

The universal oscillator model is

with $m$ mass, $\gamma$ damping coefficient, $k$ spring constant, and external forcing of angular frequency $\omega$. The natural frequency is $\omega_0=\sqrt{k/m}$.

Free oscillator ($F_0=0$). The characteristic discriminant classifies the regimes:

• Underdamped (small $\gamma$, $\Delta<0$): oscillates with amplitude decaying as $e^{-\gamma t/2m}$.
• Critical damping ($\Delta=0$): returns to equilibrium in the shortest possible time without oscillating (shock absorbers, measuring instruments).
• Overdamped ($\Delta>0$): returns slowly, without oscillating (a door with an overloaded closer).

Resonance. With zero damping and forcing at the natural frequency ($\gamma=0$, $\omega=\omega_0$), the forcing is in phase with the natural oscillation: the resonance rule gives

an amplitude growing linearly in time, up to failure. This is the phenomenon that makes a glass resonate at the right note, and that engineers avoid in bridges (Tacoma Narrows) and buildings (seismic dampers). With real damping the amplitude does not diverge but has a finite peak near $\omega_0$.

Every second-order ODE rewrites as a first-order system by setting $x_1=y$, $x_2=y'$: one gets $\mathbf{x}'=A\mathbf{x}$. More generally we study linear $2\times 2$ systems:

Eigenvalue method. Look for solutions of the form $\mathbf{x}=e^{\lambda t}\mathbf{v}$. Substituting, $\lambda\mathbf{v}=A\mathbf{v}$: i.e. $\lambda$ is an eigenvalue and $\mathbf{v}$ an eigenvector of $A$. The eigenvalues solve

With two distinct real eigenvalues $\lambda_1,\lambda_2$ and eigenvectors $\mathbf{v}_1,\mathbf{v}_2$, the general solution is the superposition of the two modes:

For complex eigenvalues $\alpha\pm i\beta$ one gets oscillating solutions $e^{\alpha t}(\cos\beta t,\sin\beta t)$; for a defective double eigenvalue a term $t\,e^{\lambda t}$ appears, as in ODEs.

The origin $\mathbf{0}$ is always an equilibrium of $\mathbf{x}'=A\mathbf{x}$. Its type and stability depend only on the eigenvalues $\lambda_{1,2}$, summarised by trace $\tau=\operatorname{tr}(A)=\lambda_1+\lambda_2$ and determinant $\delta=\det(A)=\lambda_1\lambda_2$:

• Real, same sign, negative ($\delta>0$, $\tau<0$, $\tau^2>4\delta$): stable node, all trajectories converge.
• Real, same sign, positive ($\delta>0$, $\tau>0$): unstable node, everything diverges.
• Real, opposite signs ($\delta<0$): saddle, always unstable (converges along one eigenvector, diverges along the other).
• Complex with $\operatorname{Re}<0$ ($\tau<0$, $\tau^2<4\delta$): stable focus (inward spiral).
• Complex with $\operatorname{Re}>0$: unstable focus (outward spiral).
• Purely imaginary ($\tau=0$, $\delta>0$): centre, closed orbits (undamped oscillator), neutral stability.

Practical rule. Asymptotically stable $\iff$ $\operatorname{Re}(\lambda_i)<0$ for both $\iff$ $\tau<0$ and $\delta>0$. Everything reads at a glance in the $(\tau,\delta)$ plane: the parabola $\tau^2=4\delta$ separates nodes (below) from foci (above); the half-axis $\delta<0$ is saddles; the axis $\tau=0$ with $\delta>0$ gives centres.